Between Elon Musk’s announcement of his architecture to send humans to Mars, and the community questions he answered last week, we now have a much better idea how such a trip will work. This enables intelligent discussion of many possible facets of such a mission, but the one we want to discuss today is heat management. Unlike on Earth, spacecraft are relatively thermally isolated things, losing heat only by radiation into space, and gaining heat only from sunlight.
There is also the issue of cryogenic propellant storage. Not only do the humans at the front of the craft have to kept at a comfortable temperature, the tanks of liquid oxygen and liquid methane have to be kept no warmer that their contents’ respective boiling temperatures of 90 Kelvin (-180°C, -300°F) and 110 Kelvin (-160°C, -260°F). Some of the liquid is expected to boil off during flight, but most of it is needed for the propulsive part of Mars descent (on the way back, this is a little less of a worry – if all the fuel boiled off, you could, in principle at least, use aerocapture on Earth return, and then refuel the spacecraft in Earth orbit).
There was a good discussion earlier today (on the r/SpaceX subreddit) on some of the issues surrounding thermal management. The thread uses values from NASA’s conceptual TransHab module (plus some estimation) to find the heat produced by the spaceship’s systems. However, there is an alternative, and perhaps simpler way to perform the same calculation. We know  that at Earth departure, the solar panels produce 200kW of power. If the efficiency of solar panels was constant with light intensity, we can calculate  that they would produce about 90 kW in Mars orbit. In reality, solar panels get less efficient at lower intensities, so the actual power produced will be a little lower than that. So (unless there is also a nuclear reactor that SpaceX haven’t told us about yet), all of the systems on board (regardless of number of crew) will have to use no more than 90kW of electricity between them, and correspondingly, give off no more than 90kW of heat.
What about the heat from the solar panels? A good solar panel from one of Musk’s other companies, SolarCity, would be on the order 22% efficient  at converting light into electricity, converting most of the remaining energy to heat. Admittedly, this is a very mass-produced panel, and the ones on the ITS will more than likely be more efficient than that. In fact, if you calculate their efficiency from the 200kW given by SpaceX  and their estimated area, you get 26.1% efficiency , which is very much in the same ballpark. There is also some heat absorbed by the engine block, which is the other part of the spacecraft to be in direct sunlight. If you assume little reflection (ie. most of the light becomes either heat directly, or electricity, which will become heat after it is used by the spacecraft), you get about 915 kW of heat continually absorbed by the spacecraft while still close to Earth.
There’s one other potential source of heat, which is crew. They convert energy stored in their food to heat, so they represent an additional source that the previous calculations don’t take into account. A normal human on a 2000 calorie diet gives off 100W , so a crew of 100 people would give off an additional 10 kW. This is small compared to the 915kW of solar radiation the spacecraft must lose, so we don’t need to consider it further.
Where will this heat ultimately end up? In the steady state, with the spacecraft maintaining constant temperature, there are only two places for it to go. One is out into space as radiation (good!), and the other is into the propellant tanks, boiling off the liquid methane and liquid oxygen (bad!).
The ITS has two smaller tanks (called header tanks) inside the main one which store propellant for the long haul to Mars . Thanks to our knowledge of the tank diameter  and this shot of the inside of the spacecraft , we can find the volume of the spherical liquid oxygen header tank to be about 100 cubic meters . If you put all 915kW the spacecraft was absorbing into the liquid oxygen tank, you would lose all the propellant in about 7 hours . Obviously this would be unworkable for a ~100 day transit time  that needs to use the propellant at the end of the journey to land on Mars, so to avoid this, the vast majority of the heat must be radiated into space.
Two surfaces will do most of this task: the solar panels and the hull. How much heat is radiated is a strong function of temperature (~T4), such that 10°C (18°F) warmer is about a 12% increase in heat radiated.
The temperature of the hull is constrained, though, because the same process that radiates heat into space also radiates it from the hull into the header tank. If you assume a tank that reflects 90% of the light that hits it , and that you only want to lose 10% of your fuel over 100 days, this means the hull must be kept at 127 Kelvin (-126°C, -195°F), barely warmer than the propellant in the header tank.
This would represent very little radiation into space. If all the heat must be radiated by the panels, this puts them at a temperature of 346 Kelvin (73°C, 163°F). Although this is higher than a normal panel on Earth, it’s about the temperature that ISS solar panels operate at  when they’re at their hottest.
So this all appears consistent. The vast majority of the heat will be absorbed and radiated by the solar panels – the spacecraft may need heat pumps to ensure this takes place, but doing so involves no unreasonable extremes of temperature in the panels. The tank section will have to be kept much colder (blackbody radiation hitting the header tanks is only one part of it – conduction through the structural sections will also heat the tank to some extent), though the temperatures required for passive cooling suggests that some element of active refrigeration may also be needed. All in all, from a thermal management perspective at least, Musk’s plans to explore the Red Planet seem eminently realistic and feasible.
 From the IAC presentation, at 19:51.
 A notebook containing all of the calculations for this post.
 Based on the linked press release from SolarCity.
 This is simply a units conversion.
 Slide 29
 IAC presentation, 24:43
 Slide 37
 Page 13 – peak temperature is just under 200°F (93°C, 366 K)